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%I #39 Jun 06 2023 17:16:01
%S 1,1,1,2,4,10,25,75,225,780,2704,10556,41209,178031,769129,3630780,
%T 17139600,87548580,447195609,2452523325,13450200625,78697155750,
%U 460457244900,2859220516290,17754399678409,116482516809889,764214897046969,5277304280371714
%N Number of partitions of the set {1,2,...n} having no blocks that contain both odd and even entries.
%H Alois P. Heinz, <a href="/A124419/b124419.txt">Table of n, a(n) for n = 0..500</a>
%H A. Dzhumadil’daev and D. Yeliussizov, <a href="http://arxiv.org/abs/1408.6764v1">Path decompositions of digraphs and their applications to Weyl algebra</a>, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
%H Askar Dzhumadil’daev and Damir Yeliussizov, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p10">Walks, partitions, and normal ordering</a>, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
%F a(n) = Q[n](1,1,0), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.
%F a(n) = A000110(floor(n/2)) * A000110(ceiling(n/2)). - _Alois P. Heinz_, Oct 23 2013
%F a(n) mod 2 = A088911(n). - _Alois P. Heinz_, Jun 06 2023
%e a(4) = 4 because we have 13|24, 1|24|3, 13|2|4 and 1|2|3|4.
%p Q[0]:=1: for n from 1 to 30 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 30 do Q[n]:=Q[n] od: seq(subs({t=1,s=1,x=0},Q[n]),n=0..30);
%p # second Maple program:
%p with(combinat):
%p a:= n-> bell(floor(n/2))*bell(ceil(n/2)):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Oct 23 2013
%t a[n_] := BellB[Floor[n/2]]*BellB[Ceiling[n/2]]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 20 2015, after _Alois P. Heinz_ *)
%Y Cf. A000110, A088911, A124418, A124420, A124421, A124422, A124423, A152875, A274310.
%Y Column k=0 of A124418 and of A363493.
%Y Column k=2 of A275069.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Oct 31 2006
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